Why is the price of bananas higher during certain times of the year?

It is not necessarily the case that banana prices vary in a predictable way over the course of the year. The link below from New Zealand shows that banana prices there, at least, do not vary seasonally.

If banana prices do vary, it has to be because of either supply or demand. It is possible that there might be differing levels of demand at different times of year. Consumers may prefer to eat bananas and banana products (like banana splits, perhaps) during the summer months. This would cause higher prices in the summer.

Supply factors could cause banana prices to vary as well. If more bananas are grown (world-wide) at one time of year, then that will be the time of year when prices are lowest.

So, banana prices, like all prices in a free market economy, will vary with supply and demand.